Let \(X\)  be a \(\mathrm {Poisson}(\lambda )\)  random variable. When \(\lambda \)  is an integer, \(X\)  can be seen as the sum of \(\lambda \)  independent \(\mathrm {Poisson}(1)\)  random variables. As a consequence of Cramér's Theorem (e.g., [1]}, Theorem 2.2.3), \(\lambda ^{-1} X\)  satisfies the Large Deviation Principle (LDP), namely, for any closed set \(F\subset \mathbb {R}\) ,
\(\limsup _{\lambda \rightarrow \infty }\frac{1}{\lambda }\log {\mathbb {P}}(\lambda ^{-1}X\in F)\le -\inf _{x\in F}\Lambda ^*(x),\) 
